Statistical Mechanics/Boltzmann and Gibbs factors and Partition functions/Boltzmann Factors

The first 'method of simplification' involves considering a thermal reservoir, basically a temperature bath that will keep our system of consideration at a constant temperature T.

Then by the fundamental assumption, given two energy states:



\begin{align} \frac{P(\varepsilon_1)}{P(\varepsilon_2)} & {} = \frac{g_R (U_0 - \varepsilon_1)}{g_R(U_0 - \varepsilon_2)} \\ & {} = \frac{e^{S_R(U_0-\varepsilon_1)}}{e^{S_R(U_0-\varepsilon_2)}}. \end{align} $$

Now, because of the Taylor Series, and in the presence of an infinitely large reservoir the higher-order terms vanish:



\begin{align} S_R(U_0-\varepsilon) & {} = S_R(U_0) - \varepsilon \frac{\partial S_R}{\partial U} \big|_{V,N} \\ & {} = S_R(U_0) - \frac{\varepsilon}{T}. \end{align} $$

Using this simplification we can write the previous exponential form of the ratio of probabilities:



\frac{P(\varepsilon_1)}{P(\varepsilon_2)} = \frac{e^{-\varepsilon_1 / T}}{e^{-\varepsilon_2 / T}}, $$

where $$e^{-\varepsilon / T}$$ is known as a Boltzmann factor. We will expand on its usefulness in the next section.